When a thin film of active, nematic microtubules and kinesin motor clusters is confined on the surface of a vesicle, four +1/2 topological defects oscillate in a periodic manner between tetrahedral and planar arrangements. Here a theoretical description of nematics, coupled to the relevant hydrodynamic equations, is presented here to explain the dynamics of active nematic shells. In extensile microtubule systems, the defects repel each other due to elasticity, and their collective motion leads to closed trajectories along the edges of a cube. That motion is accompanied by oscillations of their velocities, and the emergence and annihilation of vortices. When the activity increases, the system enters a chaotic regime. In contrast, for contractile systems, which are representative of some bacterial suspensions, a hitherto unknown static structure is predicted, where pairs of defects attract each other and flows arise spontaneously.

f4: Spectral analysis of defect configurations.System evolution from periodic (a,d,g, with ζ=0.0042) to quasiperiodic (b,e,h, with ζ=0.0052) and chaotic (c,f,i, with ζ=0.01). (a–c) Power spectrum of the time series of . (d–f) Projection of one defect trajectory onto the xy plane. (g–i) The four defect trajectories in three dimensions. Trajectories in g,h are made transparent and that in i are not to assist eyes.

Mentions:
We now consider how the system evolves under high activity. In Supplementary Figs 2 and 3, we show the defect trajectories for intermediate activities. As explained below, a power spectrum analysis reveals a transition from periodic to quasi-periodic and to chaotic dynamics as ζ increases34. When ζ is gradually raised to 0.005, the cube deforms and the asymmetry in the plot fades away. The above trend is discussed in Supplementary Note 3. When ζ>0.005, the plot no longer consists of a single periodic oscillation (Supplementary Fig. 3 for details). When ζ≥0.006, the defect trajectories become open (Supplementary Fig. 4). Figure 4a–c shows the power spectrum of the time series of for low (ζ=0.0042), medium (ζ=0.0052) and high activity (ζ=0.01). At ζ=0.0042, the defect trajectories in the two-dimensional and three-dimensional plots (Fig. 4d,g) are closed, and the power spectrum shows sharp peaks corresponding to their oscillation period and its harmonics. When ζ=0.0052, the defect trajectories are still closed but exhibit a more intricate geometry (Fig. 4e,h). The corresponding power spectrum exhibits equally spaced frequencies, but with significant noise in between. For ζ=0.01, the defect trajectories are chaotic outside a depletion region (Fig. 4f,i, see Supplementary Note 4 for discussion). The power spectrum shows no evidence of periodicity, and in Supplementary Fig. 4 we further show that the chaotic system is ergodic.

f4: Spectral analysis of defect configurations.System evolution from periodic (a,d,g, with ζ=0.0042) to quasiperiodic (b,e,h, with ζ=0.0052) and chaotic (c,f,i, with ζ=0.01). (a–c) Power spectrum of the time series of . (d–f) Projection of one defect trajectory onto the xy plane. (g–i) The four defect trajectories in three dimensions. Trajectories in g,h are made transparent and that in i are not to assist eyes.

Mentions:
We now consider how the system evolves under high activity. In Supplementary Figs 2 and 3, we show the defect trajectories for intermediate activities. As explained below, a power spectrum analysis reveals a transition from periodic to quasi-periodic and to chaotic dynamics as ζ increases34. When ζ is gradually raised to 0.005, the cube deforms and the asymmetry in the plot fades away. The above trend is discussed in Supplementary Note 3. When ζ>0.005, the plot no longer consists of a single periodic oscillation (Supplementary Fig. 3 for details). When ζ≥0.006, the defect trajectories become open (Supplementary Fig. 4). Figure 4a–c shows the power spectrum of the time series of for low (ζ=0.0042), medium (ζ=0.0052) and high activity (ζ=0.01). At ζ=0.0042, the defect trajectories in the two-dimensional and three-dimensional plots (Fig. 4d,g) are closed, and the power spectrum shows sharp peaks corresponding to their oscillation period and its harmonics. When ζ=0.0052, the defect trajectories are still closed but exhibit a more intricate geometry (Fig. 4e,h). The corresponding power spectrum exhibits equally spaced frequencies, but with significant noise in between. For ζ=0.01, the defect trajectories are chaotic outside a depletion region (Fig. 4f,i, see Supplementary Note 4 for discussion). The power spectrum shows no evidence of periodicity, and in Supplementary Fig. 4 we further show that the chaotic system is ergodic.

When a thin film of active, nematic microtubules and kinesin motor clusters is confined on the surface of a vesicle, four +1/2 topological defects oscillate in a periodic manner between tetrahedral and planar arrangements. Here a theoretical description of nematics, coupled to the relevant hydrodynamic equations, is presented here to explain the dynamics of active nematic shells. In extensile microtubule systems, the defects repel each other due to elasticity, and their collective motion leads to closed trajectories along the edges of a cube. That motion is accompanied by oscillations of their velocities, and the emergence and annihilation of vortices. When the activity increases, the system enters a chaotic regime. In contrast, for contractile systems, which are representative of some bacterial suspensions, a hitherto unknown static structure is predicted, where pairs of defects attract each other and flows arise spontaneously.